Connecting Inequalities to Domain and Range

There are some concepts that I know I do a great job of explaining and others that I struggle with. At the end of the school year, I made a list of things I want to do better, and a few topics from the functions unit were on there: domain and range, using the term f(x), understanding linear translations, and focusing on parent functions. As we finished up the unit on Inequalities, I knew that I wanted to start the functions unit with domain and range so that I can refer to domain and range throughout the entire unit. As I was preparing my notes, I had a "A-Ha" moment that I could present it as a compound inequality. I loved the lesson I found on compound inequalities and my students had a good understanding of writing and graphing them.

First we talked about the vocabulary for domain and range. I told them I remember that range is for y because both the g in range and y are descender, which means they dip below the writing line. I told them to pick two colored markers and color-code the words domain and range everywhere they saw them on the page. (I also made a point to tell them to pick two colors that will look different on salmon colored paper, after a student picked red and pink and couldn't tell the difference.)

The first two examples are pretty straight forward. It's when we get to the graph that things get fun. I tried the method of making the smallest possible box around the function, but several students got confused, so I changed it to finding the farthest left and right point and drawing a line to the x-axis. I then shaded the x-axis between these two values and asked, "Who can write an inequality that describes this?" I told them because this was tricky, I was offering a "Dum-Dum" to anyone will to take a risk and try it. I don't break out the candy often - it's week 7 and this was the first candy-sighting in my room, but I knew this concept was difficult. Students were fighting over trying the inequality for y.

Then I presented these three problems and asked students to talk about how they were similar and different. They pointed out that the first graph has only points, the next has endpoints, and the last has arrows, meaning it continues forever. I did the domain for the first problem and had students come up to do the range and domain and range for the middle function. For the last one, we shaded the whole x-axis and I reminded them about problems where every number is a solution and connected that idea to the domain of all real numbers. Then I asked for the maximum, which students identified at 4, so I shaded the whole y-axis below 4. Students easily saw that as y<4. Yay!

Next it was time for some scaffolded practice. I LOVE this sorting activity. I kept the cards together (they print 4 per page) and cut apart the slips for domain and range. I find that sometimes less moving pieces helps students to focus. I loved the variety of  these 20 problems and the great conversations I overheard. Students struggled most with the circles and the line, but these led to more great discussions and opportunities for me to relate them to problems they have seen. By week 7, my students are learning that I will not give them the answer, but I will ask them questions to help them come to the answer on their own [Give a man a fish, and he eats for a day. Teach a man to fish, and he will eat for a lifetime.]

As they finished, I gave them pennant problems to practice on their own. On Wednesday, we followed an early release schedule, so each student did one problem. My students on Thursday did three each. Following the gradual release, this was a great way to end because students had to come up with the domain and range on their own. And they could! They learned so much, we ran out of room on the yarn! On to more functions fun next week!

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